Nonparametric and semiparametric estimation with sequentially truncated survival data

In observational cohort studies with complex sampling schemes, truncation arises when the time to event of interest is observed only when it falls below or exceeds another random time, that is, the truncation time. In more complex settings, observation may require a particular ordering of event times; we refer to this as sequential truncation. Estimators of the event time distribution have been developed for simple left-truncated or right-truncated data. However, these estimators may be inconsistent under sequential truncation. We propose nonparametric and semiparametric maximum likelihood estimators for the distribution of the event time of interest in the presence of sequential truncation, under two truncation models. We show the equivalence of an inverse probability weighted estimator and a product limit estimator under one of these models. We study the large sample properties of the proposed estimators and derive their asymptotic variance estimators. We evaluate the proposed methods through simulation studies and apply the methods to an Alzheimer's disease study. We have developed an R package, seqTrun , for implementation of our method.     

Semiparametric analysis of survival data with left truncation and dependent right censoring

Studies of chronic life-threatening diseases often involve both mortality and morbidity. In observational studies, the data may also be subject to administrative left truncation and right censoring. Because mortality and morbidity may be correlated and mortality may censor morbidity, the Lynden-Bell estimator for left-truncated and right-censored data may be biased for estimating the marginal survival function of the non-terminal event. We propose a semiparametric estimator for this survival function based on a joint model for the two time-to-event variables, which utilizes the gamma frailty specification in the region of the observable data. First, we develop a novel estimator for the gamma frailty parameter under left truncation. Using this estimator, we then derive a closed-form estimator for the marginal distribution of the non-terminal event. The large sample properties of the estimators are established via asymptotic theory. The methodology performs well with moderate sample sizes, both in simulations and in an analysis of data from a diabetes registry.     

Model misspecification and bias for inverse probability weighting estimators of average causal effects

Commonly used semiparametric estimators of causal effects specify parametric models for the propensity score (PS) and the conditional outcome. An example is an augmented inverse probability weighting (IPW) estimator, frequently referred to as a doubly robust estimator, because it is consistent if at least one of the two models is correctly specified. However, in many observational studies, the role of the parametric models is often not to provide a representation of the data-generating process but rather to facilitate the adjustment for confounding, making the assumption of at least one true model unlikely to hold. In this paper, we propose a crude analytical approach to study the large-sample bias of estimators when the models are assumed to be approximations of the data-generating process, namely, when all models are misspecified. We apply our approach to three prototypical estimators of the average causal effect, two IPW estimators, using a misspecified PS model, and an augmented IPW (AIPW) estimator, using misspecified models for the outcome regression (OR) and the PS. For the two IPW estimators, we show that normalization, in addition to having a smaller variance, also offers some protection against bias due to model misspecification. To analyze the question of when the use of two misspecified models is better than one we derive necessary and sufficient conditions for when the AIPW estimator has a smaller bias than a simple IPW estimator and when it has a smaller bias than an IPW estimator with normalized weights. If the misspecification of the outcome model is moderate, the comparisons of the biases of the IPW and AIPW estimators show that the AIPW estimator has a smaller bias than the IPW estimators. However, all biases include a scaling with the PS-model error and we suggest caution in modeling the PS whenever such a model is involved. For numerical and finite sample illustrations, we include three simulation studies and corresponding approximations of the large-sample biases. In a dataset from the National Health and Nutrition Examination Survey, we estimate the effect of smoking on blood lead levels.     

Combining multiple imputation and inverse-probability weighting

Two approaches commonly used to deal with missing data are multiple imputation (MI) and inverse-probability weighting (IPW). IPW is also used to adjust for unequal sampling fractions. MI is generally more efficient than IPW but more complex. Whereas IPW requires only a model for the probability that an individual has complete data (a univariate outcome), MI needs a model for the joint distribution of the missing data (a multivariate outcome) given the observed data. Inadequacies in either model may lead to important bias if large amounts of data are missing. A third approach combines MI and IPW to give a doubly robust estimator. A fourth approach (IPW/MI) combines MI and IPW but, unlike doubly robust methods, imputes only isolated missing values and uses weights to account for remaining larger blocks of unimputed missing data, such as would arise, e.g., in a cohort study subject to sample attrition, and/or unequal sampling fractions. In this article, we examine the performance, in terms of bias and efficiency, of IPW/MI relative to MI and IPW alone and investigate whether the Rubin's rules variance estimator is valid for IPW/MI. We prove that the Rubin's rules variance estimator is valid for IPW/MI for linear regression with an imputed outcome, we present simulations supporting the use of this variance estimator in more general settings, and we demonstrate that IPW/MI can have advantages over alternatives. IPW/MI is applied to data from the National Child Development Study.     

Approximate nonparametric corrected-score method for joint modeling of survival and longitudinal data measured with error

We consider the problem of jointly modeling survival time and longitudinal data subject to measurement error. The survival times are modeled through the proportional hazards model and a random effects model is assumed for the longitudinal covariate process. Under this framework, we propose an approximate nonparametric corrected-score estimator for the parameter, which describes the association between the time-to-event and the longitudinal covariate. The term nonparametric refers to the fact that assumptions regarding the distribution of the random effects and that of the measurement error are unnecessary. The finite sample size performance of the approximate nonparametric corrected-score estimator is examined through simulation studies and its asymptotic properties are also developed. Furthermore, the proposed estimator and some existing estimators are applied to real data from an AIDS clinical trial.     

Improved Horvitz–Thompson Estimation of Model Parameters from Two-phase Stratified Samples: Applications in Epidemiology

The case-cohort study involves two-phase samplings: simple random sampling from an infinite superpopulation at phase one and stratified random sampling from a finite cohort at phase two. Standard analyses of case-cohort data involve solution of inverse probability weighted (IPW) estimating equations, with weights determined by the known phase two sampling fractions. The variance of parameter estimates in (semi)parametric models, including the Cox model, is the sum of two terms: (i) the model-based variance of the usual estimates that would be calculated if full data were available for the entire cohort; and (ii) the design-based variance from IPW estimation of the unknown cohort total of the efficient influence function (IF) contributions. This second variance component may be reduced by adjusting the sampling weights, either by calibration to known cohort totals of auxiliary variables correlated with the IF contributions or by their estimation using these same auxiliary variables. Both adjustment methods are implemented in the R survey package. We derive the limit laws of coefficients estimated using adjusted weights. The asymptotic results suggest practical methods for construction of auxiliary variables that are evaluated by simulation of case-cohort samples from the National Wilms Tumor Study and by log-linear modeling of case-cohort data from the Atherosclerosis Risk in Communities Study. Although not semiparametric efficient, estimators based on adjusted weights may come close to achieving full efficiency within the class of augmented IPW estimators.     

Estimation of a Concordance Probability for Doubly Censored Time-to-Event Data

Evaluating the relationship between a response variable and explanatory variables is important to establish better statistical models. Concordance probability is one measure of this relationship and is often used in biomedical research. Concordance probability can be seen as an extension of the area under the receiver operating characteristic curve. In this study, we propose estimators of concordance probability for time-to-event data subject to double censoring. A doubly censored time-to-event response is observed when either left or right censoring may occur. In the presence of double censoring, existing estimators of concordance probability lack desirable properties such as consistency and asymptotic normality. The proposed estimators consist of estimators of the left-censoring and the right-censoring distributions as a weight for each pair of cases, and reduce to the existing estimators in special cases. We show the statistical properties of the proposed estimators and evaluate their performance via numerical experiments.     

Confidence Intervals of the Attributable Risk under Cross‐Sectional Sampling with Confounders

Since it can account for both the strength of the association between exposure to a risk factor and the underlying disease of interest and the prevalence of the risk factor, the attributable risk (AR) is probably the most commonly used epidemiologic measure for public health administrators to locate important risk factors. This paper discusses interval estimation of the AR in the presence of confounders under cross‐sectional sampling. This paper considers four asymptotic interval estimators which are direct generalizations of those originally proposed for the case of no confounders, and employs Monte Carlo simulation to evaluate the finite‐sample performance of these estimators in a variety of situations. This paper finds that interval estimators using Wald's test statistic and a quadratic equation suggested here can consistently perform reasonably well with respect to the coverage probability in all the situations considered here. This paper notes that the interval estimator using the logarithmic transformation, that is previously found to consistently perform well for the case of no confounders, may have the coverage probability less than the desired confidence level when the underlying common prevalence rate ratio (RR) across strata between the exposure and the non‐exposure is large (≥4). This paper further notes that the interval estimator using the logit transformation is inappropriate for use when the underlying common RR ≐ 1. On the other hand, when the underlying common RR is large (≥4), this interval estimator is probably preferable to all the other three estimators. When the sample size is large (≥400) and the RR ≥ 2 in the situations considered here, this paper finds that all the four interval estimators developed here are essentially equivalent with respect to both the coverage probability and the average length.     

Clustered restricted mean survival time regression

For multicenter randomized trials or multilevel observational studies, the Cox regression model has long been the primary approach to study the effects of covariates on time-to-event outcomes. A critical assumption of the Cox model is the proportionality of the hazard functions for modeled covariates, violations of which can result in ambiguous interpretations of the hazard ratio estimates. To address this issue, the restricted mean survival time (RMST), defined as the mean survival time up to a fixed time in a target population, has been recommended as a model-free target parameter. In this article, we generalize the RMST regression model to clustered data by directly modeling the RMST as a continuous function of restriction times with covariates while properly accounting for within-cluster correlations to achieve valid inference. The proposed method estimates regression coefficients via weighted generalized estimating equations, coupled with a cluster-robust sandwich variance estimator to achieve asymptotically valid inference with a sufficient number of clusters. In small-sample scenarios where a limited number of clusters are available, however, the proposed sandwich variance estimator can exhibit negative bias in capturing the variability of regression coefficient estimates. To overcome this limitation, we further propose and examine bias-corrected sandwich variance estimators to reduce the negative bias of the cluster-robust sandwich variance estimator. We study the finite-sample operating characteristics of proposed methods through simulations and reanalyze two multicenter randomized trials.     

Estimation for the optimal combination of markers without modeling the censoring distribution

Summary .  In the time-dependent receiver operating characteristic curve analysis with several baseline markers, research interest focuses on seeking appropriate composite markers to enhance the accuracy in predicting the vital status of individuals over time. Based on censored survival data, we proposed a more flexible estimation procedure for the optimal combination of markers under the validity of a time-varying coefficient generalized linear model for the event time without restrictive assumptions on the censoring pattern. The consistency of the proposed estimators is also established in this article. In contrast, the inverse probability weighting (IPW) approach might introduce a bias when the selection probabilities are misspecified in the estimating equations. The performance of both estimation procedures are examined and compared through a class of simulations. It is found from the simulation study that the proposed estimators are far superior to the IPW ones. Applying these methods to an angiography cohort, our estimation procedure is shown to be useful in predicting the time to all-cause and coronary artery disease related death.     

Estimation of Heritability in Heteroscedastic One‐Way Unbalanced Random Model

A modified estimator of heritability is proposed under heteroscedastic one way unbalanced random model. The distribution, moments and probability of permissible values (PPV) for conventional and modified estimators are derived. The behaviour of two estimators has been investigated, numerically, to devise a suitable estimator of heritability under variance heterogeneity. The numerical results reveal that under balanced case the heteroscedasticity affects the bias, MSE and PPV of conventional estimator, marginally. In case of unbalanced situations, the conventional estimator underestimates the parameter when more variable group has more observations and overestimates when more variable group has less observations, MSE of the conventional estimator decreases when more variable group has more observations and increases when more variable group has less observations and PPV is marginally decreased. The MSE and PPV are comparable for two estimators while the bias of modified estimator is less than the conventional estimator particularly for small and medium values of the parameter. These results suggest the use of modified estimator with equal or more observations for more variable group in presence of variance heterogeneity.     

Nonparametric estimation of a Markov 'illness-death' process from interval-censored observations, with application to diabetes survival data

The nonparametric estimation of the cumulative transition intensityfunctions in a threestate time-nonhomogeneous Markov processwith irreversible transitions, an ‘illness-death’model, is considered when times of the intermediate transition,e.g. onset of a disease, are interval-censored. The times of‘death’ are assumed to be known exactly or to beright-censored. In addition the observed process may be left-truncated.Data of this type arise when the process is sampled periodically.For example, when the patients are monitored through periodicexaminations the observations on times of change in their diseasestatus will be interval-censored. Under the sampling schemeconsidered here the Nelson–Aalen estimator (Aalen, 1978)for a cumulative transition intensity is not applicable. Inthe proposed method the maximum likelihood estimators of someof the transition intensities are derived from the estimatorsof the corresponding subdistribution functions. The maximumlikelihood estimators are shown to have a self-consistency property.The self-consistency algorithm is developed for the computationof the estimators. This approach generalises the results fromTurnbull (1976) and Frydman (1992). The methods are illustratedwith diabetes survival data.     

Interval mapping of quantitative trait loci for time-to-event data with the proportional hazards mixture cure model

Interval mapping using normal mixture models has been an important tool for analyzing quantitative traits in experimental organisms. When the primary phenotype is time-to-event, it is natural to use survival models such as Cox's proportional hazards model instead of normal mixtures to model the phenotype distribution. An extra challenge for modeling time-to-event data is that the underlying population may consist of susceptible and nonsusceptible subjects. In this article, we propose a semiparametric proportional hazards mixture cure model which allows missing covariates. We discuss applications to quantitative trait loci (QTL) mapping when the primary trait is time-to-event from a population of mixed susceptibility. This model can be used to characterize QTL effects on both susceptibility and time-to-event distribution, and to estimate QTL location. The model can naturally incorporate covariate effects of other risk factors. Maximum likelihood estimates for the parameters in the model as well as their corresponding variance estimates can be obtained numerically using an EM-type algorithm. The proposed methods are assessed by simulations under practical settings and illustrated using a real data set containing survival times of mice after infection with Listeria monocytogenes. An extension to multiple intervals is also discussed.     

A Monte Carlo evaluation of five interval estimators for the relative risk in sparse data

The relative risk (RR) is one of the most frequently used indices to measure the strength of association between a disease and a risk factor in etiological studies or the efficacy of an experimental treatment in clinical trials. In this paper, we concentrate attention on interval estimation of RR for sparse data, in which we have only a few patients per stratum, but a moderate or large number of strata. We consider five asymptotic interval estimators for RR, including a weighted least-squares (WLS) interval estimator with an ad hoc adjustment procedure for sparse data, an interval estimator proposed elsewhere for rare events, an interval estimator based on the Mantel-Haenszel (MH) estimator with a logarithmic transformation, an interval estimator calculated from a quadratic equation, and an interval estimator derived from the ratio estimator with a logarithmic transformation. On the basis of Monte Carlo simulations, we evaluate and compare the performance of these five interval estimators in a variety of situations. We note that, except for the cases in which the underlying common RR across strata is around 1, using the WLS interval estimator with the adjustment procedure for sparse data can be misleading. We note further that using the interval estimator suggested elsewhere for rare events tends to be conservative and hence leads to loss of efficiency. We find that the other three interval estimators can consistently perform well even when the mean number of patients for a given treatment is approximately 3 patients per stratum and the number of strata is as small as 20. Finally, we use a mortality data set comparing two chemotherapy treatments in patients with multiple myeloma to illustrate the use of the estimators discussed in this paper.     

Estimation of separable direct and indirect effects in continuous time

Many research questions involve time-to-event outcomes that can be prevented from occurring due to competing events. In these settings, we must be careful about the causal interpretation of classical statistical estimands. In particular, estimands on the hazard scale, such as ratios of cause-specific or subdistribution hazards, are fundamentally hard to interpret causally. Estimands on the risk scale, such as contrasts of cumulative incidence functions, do have a clear causal interpretation, but they only capture the total effect of the treatment on the event of interest; that is, effects both through and outside of the competing event. To disentangle causal treatment effects on the event of interest and competing events, the separable direct and indirect effects were recently introduced. Here we provide new results on the estimation of direct and indirect separable effects in continuous time. In particular, we derive the nonparametric influence function in continuous time and use it to construct an estimator that has certain robustness properties. We also propose a simple estimator based on semiparametric models for the two cause-specific hazard functions. We describe the asymptotic properties of these estimators and present results from simulation studies, suggesting that the estimators behave satisfactorily in finite samples. Finally, we reanalyze the prostate cancer trial from Stensrud et al. (2020).     

Exploiting nonsystematic covariate monitoring to broaden the scope of evidence about the causal effects of adaptive treatment strategies

In studies based on electronic health records (EHR), the frequency of covariate monitoring can vary by covariate type, across patients, and over time, which can limit the generalizability of inferences about the effects of adaptive treatment strategies. In addition, monitoring is a health intervention in itself with costs and benefits, and stakeholders may be interested in the effect of monitoring when adopting adaptive treatment strategies. This paper demonstrates how to exploit nonsystematic covariate monitoring in EHR‐based studies to both improve the generalizability of causal inferences and to evaluate the health impact of monitoring when evaluating adaptive treatment strategies. Using a real world, EHR‐based, comparative effectiveness research (CER) study of patients with type II diabetes mellitus, we illustrate how the evaluation of joint dynamic treatment and static monitoring interventions can improve CER evidence and describe two alternate estimation approaches based on inverse probability weighting (IPW). First, we demonstrate the poor performance of the standard estimator of the effects of joint treatment‐monitoring interventions, due to a large decrease in data support and concerns over finite‐sample bias from near‐violations of the positivity assumption (PA) for the monitoring process. Second, we detail an alternate IPW estimator using a no direct effect assumption. We demonstrate that this estimator can improve efficiency but at the potential cost of increase in bias from violations of the PA for the treatment process.     

K‐Sample comparisons using propensity analysis

In this paper, we investigate K‐group comparisons on survival endpoints for observational studies. In clinical databases for observational studies, treatment for patients are chosen with probabilities varying depending on their baseline characteristics. This often results in noncomparable treatment groups because of imbalance in baseline characteristics of patients among treatment groups. In order to overcome this issue, we conduct propensity analysis and match the subjects with similar propensity scores across treatment groups or compare weighted group means (or weighted survival curves for censored outcome variables) using the inverse probability weighting (IPW). To this end, multinomial logistic regression has been a popular propensity analysis method to estimate the weights. We propose to use decision tree method as an alternative propensity analysis due to its simplicity and robustness. We also propose IPW rank statistics, called Dunnett‐type test and ANOVA‐type test, to compare 3 or more treatment groups on survival endpoints. Using simulations, we evaluate the finite sample performance of the weighted rank statistics combined with these propensity analysis methods. We demonstrate these methods with a real data example. The IPW method also allows us for unbiased estimation of population parameters of each treatment group. In this paper, we limit our discussions to survival outcomes, but all the methods can be easily modified for any type of outcomes, such as binary or continuous variables.     

On estimation in relative survival

Estimation of relative survival has become the first and the most basic step when reporting cancer survival statistics. Standard estimators are in routine use by all cancer registries. However, it has been recently noted that these estimators do not provide information on cancer mortality that is independent of the national general population mortality. Thus they are not suitable for comparison between countries. Furthermore, the commonly used interpretation of the relative survival curve is vague and misleading. The present article attempts to remedy these basic problems. The population quantities of the traditional estimators are carefully described and their interpretation discussed. We then propose a new estimator of net survival probability that enables the desired comparability between countries. The new estimator requires no modeling and is accompanied with a straightforward variance estimate. The methods are described on real as well as simulated data.     

Analysis of longitudinal marginal structural models

In this article we construct and study estimators of the causal effect of a time-dependent treatment on survival in longitudinal studies. We employ a particular marginal structural model (MSM), proposed by Robins (2000), and follow a general methodology for constructing estimating functions in censored data models. The inverse probability of treatment weighted (IPTW) estimator of Robins et al. (2000) is used as an initial estimator and forms the basis for an improved, one-step estimator that is consistent and asymptotically linear when the treatment mechanism is consistently estimated. We extend these methods to handle informative censoring. The proposed methodology is employed to estimate the causal effect of exercise on mortality in a longitudinal study of seniors in Sonoma County. A simulation study demonstrates the bias of naive estimators in the presence of time-dependent confounders and also shows the efficiency gain of the IPTW estimator, even in the absence such confounding. The efficiency gain of the improved, one-step estimator is demonstrated through simulation.     

A regression-based method for estimating mean treatment cost in the presence of right-censoring

In many clinical trials and evaluations using medical care administrative databases it is of interest to estimate not only the survival time of a given treatment modality but also the total associated cost. The most widely used estimator for data subject to censoring is the Kaplan-Meier (KM) or product-limit (PL) estimator. The optimality properties of this estimator applied to time-to-event data (consistency, etc.) under the assumptions of random censorship have been established. However, whenever the relationship between cost and survival time includes an error term to account for random differences among patients' costs, the dependency between cumulative treatment cost at the time of censoring and at the survival time results in KM giving biased estimates. A similar phenomenon has previously been noted in the context of estimating quality-adjusted survival time. We propose an estimator for mean cost which exploits the underlying relationship between total treatment cost and survival time. The proposed method utilizes either parametric or nonparametric regression to estimate this relationship and is consistent when this relationship is consistently estimated. We then present simulation results which illustrate the gain in finite-sample efficiency when compared with another recently proposed estimator. The methods are then applied to the estimation of mean cost for two studies where right-censoring was present. The first is the heart failure clinical trial Studies of Left Ventricular Dysfunction (SOLVD). The second is a Health Maintenance Organization (HMO) database study of the cost of ulcer treatment.